Cumulants and large deviations for the linear statistics of the one-dimensional trapped Riesz gas

TL;DR

This paper analyzes the fluctuations and large deviations of linear statistics in the one-dimensional trapped Riesz gas, deriving explicit formulas for cumulants across different interaction regimes and revealing an evaporation transition in the fluctuation behavior.

Contribution

It provides the first comprehensive analytic formulas for cumulants and large deviation properties of linear statistics in the Riesz gas for various interaction ranges and functions.

Findings

Explicit cumulant formulas for general $k>-2$

Identification of an evaporation transition in large deviations

Higher order cumulants for full counting statistics near the edge

Abstract

We consider the classical trapped Riesz gas, i.e., $N$ particles at positions $x_i$ in one dimension with a repulsive power law interacting potential $\propto 1/|x_i-x_j|^{k}$, with $k>-2$, in an external confining potential of the form $V(x) \sim |x|^n$. We focus on the equilibrium Gibbs state of the gas, for which the density has a finite support $[-\ell_0/2,\ell_0/2]$. We study the fluctuations of the linear statistics ${\cal L}_N = \sum_{i=1}^N f(x_i)$ in the large $N$ limit for smooth functions $f(x)$. We obtain analytic formulae for the cumulants of ${\cal L}_N$ for general $k>-2$. For long range interactions, i.e. $k<1$, which include the log-gas ($k \to 0$) and the Coulomb gas ($k =-1$) these are obtained for monomials $f(x)= |x|^m$. For short range interactions, i.e. $k>1$, which include the Calogero-Moser model, i.e. $k=2$, we compute the third cumulant of ${\cal L}_N$ for general $f(x)$ and arbitrary cumulants for monomials $f(x)= |x|^m$. We also obtain the large deviation form of the probability distribution of ${\cal L}_N$, which exhibits an "evaporation transition" where the fluctuation of ${\cal L}_N$ is dominated by the one of the largest $x_i$. In addition, in the short range case, we extend our results to a (non-smooth) indicator function $f(x)$, obtaining thereby the higher order cumulants for the full counting statistics of the number of particles in an interval $[-L/2,L/2]$. We show in particular that they exhibit an interesting scaling form as $L/2$ approaches the edge of the gas $L/\ell_0 \to 1$, which we relate to the large deviations of the emptiness probability of the complementary interval on the real line.